Achieving the Age-Energy Tradeoff with a Finite-Battery Energy Harvesting Source

We study the problem of minimizing the time-average expected Age of Information for status updates sent by an energy-harvesting source with a finite-capacity battery. In prior literature, optimal policies were observed to have a threshold structure under Poisson energy arrivals, for the special case of a unit-capacity battery. In this paper, we generalize this result to any (integer) battery capacity, and explicitly characterize the threshold structure. We obtain tools to derive the optimal policy for arbitrary energy buffer (i.e. battery) size. One of these results is the unexpected equivalence of the minimum average AoI and the optimal threshold for the highest energy state.

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I Introduction

The Age of Information (AoI) was proposed in [1, 2] as a performance metric that measures the freshness of information in status-update systems. For a flow of information updates sent from a source to a destination, status age is defined as the time elapsed since the newest update available was generated at the source. That is, if is the largest among the time-stamps of all packets received by time , status age is defined as:

(1)

The Age of Information (AoI) usually refers to the time-average of . AoI is a particularly relevant performance metric for status-update applications that have growing importance in social networks, remote monitoring [3, 4], machine-type communication (smart cities, industrial manufacturing, telerobotics, IoT).

AoI was analyzed under various queueing system models, service disciplines and queue management policies in recent literature (e.g., [5, 6, 7, 8, 9, 10, 11, 12]. The control and optimization of AoI for an active source that can generate updates at will, was studied in [13, 14].

The relation of energy and AoI was studied as early as 2015: The problem of AoI-optimal generation of status updates when the source is constrained by an arbitrary sequence of energy arrivals was formulated in [15], resulting in the optimal offline solution and an online policy. The study in [16] considered the optimization of AoI under a long-term average rate of energy harvesting, when update transmissions are subject to random delays in the network. Both studies observed that AoI-optimal policies tend to be lazy, in the sense that they may intentionally impose a waiting time before sending the next update. That is, for maximum freshness, one may sometimes send updates at a rate lower than one is allowed to- which may be counter-intuitive at first sight.

The online problem in [15] was extended to a continuous-time formulation with Poisson energy arrivals, finite energy storage (battery) capacity, and random packet errors in the channel in [17]

. An age-optimal threshold policy was proposed for the unit battery case, and the achievable AoI for arbitrary battery size was bounded for a channel with a constant error packet error probability. Optimal threshold policies for the unit battery and infinite battery capacity cases were found for a channel with no errors, in the concurrent study in 

[18]. The problem of characterizing optimal policies for arbitrary battery sizes remained open.

In [19], the offline results in [17] were extended considering fixed non-zero service time and the result is used to obtain a solution for the two-hop scenario. Another offline problem under energy harvesting was investigated in [20] where the transmission delay of an update is controlled by energy consumed on its transmission.

This paper extends [17], making the following contributions:

  • A more general description of the policy space for age-optimal scheduling, including threshold policies with age-based thresholds that are monotone in energy state, is formulated.

  • Following the study in [17], it was conjectured that for any battery size, the optimal threshold on the age for the highest energy state is actually equal to the minimum AoI. This conjecture is proved to be correct.

  • Optimal thresholds are obtained numerically for integer battery size up to 5.

Ii System Model

Consider an energy harvesting transmitter that sends update packets to a destination, as illustrated in Fig 1. Suppose that the transmitter has a finite battery which is capable of storing up to units of energy. Transmission of each update packet consumes a unit of energy. Let denote the amount of energy stored in the battery at time . The timing of status updates are controlled by a sampler which can monitor the battery level at all time .

Fig. 1: System Model.

We assume that when an update is given to the transmitter, it is instantaneously transmitted 111This corresponds to an assumption of instantaneous service, i.e, the duration of packet transmission is ignored. This is an appropriate model for sporadic transmissions (e.g., a sensor reporting temperature) where the time between two updates is typically much larger than a packet transmission duration..

Let and denote the number of energy units that have arrived and the number of updates that have sent out by time , respectively. We assume that the energy arrival process is Poisson with a rate . Energy arriving while the battery is full is lost (cannot be stored or used).

The system starts to operate at time . Let denote the generation time of the -th update packet such that . An update policy is defined by a sequence of update instants . In many status-update systems (e.g., a sensor reporting temperature ), the update packets are only sent out sporadically and the packet size is quite small. Hence, the duration for transmitting a packet is much smaller than the difference between two subsequent update times. Motivated by this, we assume that the packet transmission time can be approximated as zero. With this assumption, the age at a status generation is zero, i.e. for any , and the age at any time is:

(2)

The battery level before the -st update instant is given by the following:

(3)

We first define the set of energy-causal update policies:

Definition 1.

A policy is said to be energy-causal if no update packet is sent out when the battery is empty, i.e., for all .

The information available up to some time is represented by which is the -field generated by the sequence of energy arrivals and updates, i.e., . The set of online update policies is defined as follows:

Definition 2.

An energy-causal policy is said to be online if no update instant is determined based on future information, i.e., does not depend on future events, i.e., for all and .

Let denote the set of online update policies. The time-average expected age can be expressed as:

(4)

Let represent the inter-update duration between updates and , i.e., . Then, the time-average expected age in (4) can be equivalently expressed as:

(5)

The goal of this paper is to find the optimal update policy for minimizing the time-average expected age, which is formulated as:

(6)

Iii Main Results

We begin with a result guaranteeing the existence of threshold-type policies that are optimal. We define such policies as follows:

Definition 3.

An online policy is said to be a threshold policy if:

(7)

where denotes the threshold for sending an update when the battery level is for .

Let be the set of threshold policies. First, we note the following:

Theorem 1.

There exists a threshold policy that solves (6).

In our search for an optimal policy, we can reduce the space of policies further,

Definition 4.

A threshold policy is said to be a monotone threshold policy if .

Let be the set of monotone threshold policies. The following is true:

Theorem 2.

There exists a monotone threshold policy that solves (6).

Theorem 2 implies that in the optimal update policy, update packets are sent out more frequently when the battery level is high.

Fig. 2: An illustration of the state space and transitions for a policy in .

To understand the time evolution of and for policies in , consider the illustration in Fig. 2. It can be seen from Fig. 2 that when , the next update of a policy occurs before than some time if and only if there occur energy arrivals before than . Accordingly, for policies in

, the cumulative distribution function (CDF) of inter-update durations,

can be expressed as:

(8)

where obeys the Erlang distribution at rate with parameter , for , and for . From (8), an expression for the transition probability for can be derived:

(9)

Hence, energy states sampled at update instants can be described as a DTMC with the transition probabilities in (9). When thresholds are finite, this DTMC is ergodic as any energy state is reachable from any other energy state with positive probability in steps.

Next, we show the main structural result satisfied by the thresholds of any optimal policy in .

Theorem 3.

An optimal policy for solving (6) is a monotone threshold policy that satisfies the following property: The threshold for sending an update packet when the battery is full is equal to the minimum time-average expected age, i.e.,

(10)

This follows from the following two results:

Lemma 1.

Consider non-negative random variable

, if:

where and is the CDF of a non-negative random variable for every , then:

Corollary 1.

The inter-update intervals, , for any satisfy the following:

(11)

Note that the transition probabilities (9) do not depend on hence the steady-state probabilities obtained from (9) also do not depend on . This leads to a property of which is shown in Theorem 3. The unit-battery case , i.e., case was solved in  [18] and  [17], hence we skip the case and continue with the case where we can show the result below:

Theorem 4.

When , the average age can be expressed as:

(12)

where

and

Iv Numerical Results

For battery sizes , the policies in are numerically optimized giving AoI versus energy arrival rate (Poisson) curves in Fig 3.

Y[t] X[b] a[l] b[l] c[l] d[l] e[l] f[l]

Fig. 3: AoI versus energy arrival rate (Poisson) for different battery sizes .
0.90 - - - - 0.90
1.5 0.72 - - - 0.72
1.5 1.2 0.64 - - 0.64
1.5 1.2 0.96 0.604 - 0.604
1.5 1.2 0.96 0.9 0.582 0.582
TABLE I: Optimal thresholds for different battery sizes for

V Conclusion

This paper explored the age-energy tradeoff for status updates sent by a finite-battery source that is charged intermittently by Poisson energy arrivals. The objective was to design a policy for the source to send updates to minimizing average status age using the given energy harvests, known and used in an online manner. A threshold policy is one that transmits when age exceeds a particular threshold for any battery state. It is shown that there is an online energy-causal threshold policy with monotone thresholds that optimally solves the problem. In particular, the smallest of the thresholds, the one used when the battery is full, has a value that matches the optimal average age.

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